Question

A piecewise function is defined as follows: for all rational numbers, f(x) = 1, for all irrational numbers, f(x) = 0. will the resulting graph be a line on f(x) = 0 with discontinuities? a line on f(x) = 1 with discontinuities, or a scattering of individual points?

Answer 1

i tend to agree with line on y=0 with discontinuities, there are far more irrational numbers than rational when looking at all real numbers

Answer 2

is that because they are uncountable?

Answer 3

kind of , rational numbers are uncountable too but for every gap between 2 rational numbers there could be an infinite number of irrational numbers

Answer 4

well you will "see" two lines , y = 1 and y = 0

Answer 5

it doesnt matter there are more irrationals than rational, so what

Answer 6

Are there more irrational numbers than rational is really the question

Answer 7

as far as you are concerned, between any two irrationals is a rational, between any two rationals is an irrational, etc

Answer 8

or more generally, pick two points, between them is an irrational and a rational

Answer 9

pick 2 distinct points

Answer 10

so it is a scattering of individual points, neither line exists

Answer 11

very true, imagine we had a microscope ;)

Answer 12

right, its not a line ,. but at the macro level it "looks" like a line

Answer 13

So the argument I have heard that because irrational numbers are uncountable, it is a "larger" infinity than the countable set of rational numbers, is false - or that there is only one more

Answer 14

that is a true statement, but that doesnt mean that you can order them , rational, irrational, rational, irrational, etc

Answer 15

on the number line. so ...

Answer 16

it gets even weirder. between any 2 rationals is an irrational. between any two irrationals is a rational. so where exactly are all the "extra" irrationals ?

Answer 17

between any 2 rational numbers there exists an irrational number. between any 2 irrational numbers there exists a rational number. so you might think, the number of rational and irrational should be about the same, since you can keep going with this statement

Answer 18

okay, you said, "that is a true statement, but that doesnt mean that you can order them , rational, irrational, rational, irrational, etc" does that mean that there can and cannot be two irrational numbers without a rational between them?

Answer 19

there cannot be two distinct irrationals without a rational between them, correct

Answer 20

so its kind of paradoxical

Answer 21

Do you have a good source where I could read more? Wiki gets too techie too quickly

Answer 22

yeah, thats true. well you can google cantor's infinity

Answer 23

but this question is not addressed specifically, its something that my math professor told me

Answer 24

there are many layman easy websites

Answer 25

Yeah, I want something between the two levels

Answer 26

yeah i know the feeling

Answer 27

Also, I think it's neat to see hebrew letters come into math for this question!

Answer 28

well college libraries have some gradual immersion books

Answer 29

yeah, thats cool

Answer 30

the big question in advanced math, does there exist a cardinality between aleph null ( natural numbers) and the cardinality of the real numbers

Answer 31

it can be shown that aleph null is the "least" infinity , as weird as that sounds

Answer 32

and we can make a hierarchy of infinite cardinalities. use the natural number cardinality, and take the cardinality of the power set of the natural numbers

Answer 33

so we start with | N | , where N = { 1,2,3,...}

Answer 34

ok

Answer 35

where |N | means cardinality , that means the size of the set

Answer 36

right

Answer 37

now we know for any set , |P (S)| > |S|

Answer 38

P(S) is the power set of S

Answer 39

only for finite sets?

Answer 40

for any set

Answer 41

ok

Answer 42

it comes in handy for infinite sets

Answer 43

what a power set ?

Answer 44

2^S, right?

Answer 45

the number of permutations within the set

Answer 46

right, thats the shorthand for it

Answer 47

well, its the set of all subsets of the set

Answer 48

i dont know about permutations, hmmm

Answer 49

I thought infinity was defined as the set that has the same cardinality as any of its proper subsets

Answer 50

yes thats a definition of an infinite set

Answer 51

but im talking about power set, what is the power set operation

Answer 52

ok say you have { 1, 2 } , the power set is
{ {}, {1}, {2} , {1,2} }

Answer 53

S= { 1, 2 } , then P(S)= { {}, {1}, {2} , {1,2} }

Answer 54

right

Answer 55

now cantor proved a delightful theorem , that
| P ( S) | > | S| for any set S

Answer 56

so the powerset of integers is larger I mean has a higher cardinality than the set of integers

Answer 57

exactly

Answer 58

ok so

Answer 59

| P ( N ) | > | N | , correct ?

Answer 60

I sense a trick coming up...

Answer 61

but, yes

Answer 62

http://en.wikipedia.org/wiki/Cardinality_of_the_continuum

Answer 63

ok it turns out that |R| which we use shorthand c , it turns out
|R| = | P ( N ) | ,

Answer 64

now the question is, is there an infinity between |N| and | P ( N ) |

Answer 65

is the real numbers the smallest infinity after the natural numbers

Answer 66

Wouldn't rational and irrational be smaller than real?

Answer 67

we have a hierarchy of infinities,
|N| < |P(N)| < |P (P(N))| < ...

Answer 68

oh boy, rational is the same cardinality as natural numbers

Answer 69

the irrationals are uncountable

Answer 70

So cantor's work doesn't cover uncountable sets?

Answer 71

I find it hard to assign the same cardinality to rationals as naturals, which means I only have a working understanding of cardinality and not a real grasp on it.

Answer 72

help